So far, we have provided all the necessary ingredients to discuss the (spherical) unitary representation theory for \(G = \varinjlim G_n\) . As a consequence, we saw that the study of (spherical) unitary representations comes down to that of locally normal \((\vartheta ^t,-1)\) -KMS states. Thus, one may regard such a state as a quantum analog of character, and this idea was indeed employed by Sato in [1]} (and his later works) to develop the asymptotic representation theory for quantum groups (without appealing to spherical representations). In our opinion, it is more appropriate to formulate quantized characters as `functions over \(G\) '. This aspect was already considered by Sato [1]} based on Arano's suggestion to him. However, Sato's discussion there is not entirely satisfactory (at least to us), because his discussion depends on Gorin's work [3]} and is performed only in a special case. Thus, we give here a new formulation of quantized characters of \(G\) , which enables us to give a (hopefully nice) interpretation to Gorin's work in terms of quantum groups in the next subsection.
